Finite Math Examples

$7 y^{2} = 6 y^{2} - 6 y + 3$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Subtract $6 y^{2}$ from both sides of the equation.

$7 y^{2} - 6 y^{2} = - 6 y + 3$

Step 1.1.2

Add $6 y$ to both sides of the equation.

$7 y^{2} - 6 y^{2} + 6 y = 3$

Step 1.1.3

Subtract $3$ from both sides of the equation.

$7 y^{2} - 6 y^{2} + 6 y - 3 = 0$

Step 1.2

Subtract $6 y^{2}$ from $7 y^{2}$ .

$y^{2} + 6 y - 3 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = 6$ , and $c = - 3$ into the quadratic formula and solve for $y$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 3)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 3$ .

$y = \frac{- 6 \pm \sqrt{36 + 12}}{2 \cdot 1}$

Step 4.1.3

Add $36$ and $12$ .

$y = \frac{- 6 \pm \sqrt{48}}{2 \cdot 1}$

Step 4.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 4.1.4.1

Factor $16$ out of $48$ .

$y = \frac{- 6 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 4.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 6 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$y = \frac{- 6 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{- 6 \pm 4 \sqrt{3}}{2}$

Step 4.3

Simplify $\frac{- 6 \pm 4 \sqrt{3}}{2}$ .

$y = - 3 \pm 2 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - 3 \pm 2 \sqrt{3}$

Decimal Form:

$y = 0.46410161 \dots, - 6.46410161 \dots$